Georgia Institute of Technology College of Sciences

Graphs are central objects of study in Discrete Mathematics. A graph consists of a set of vertices, some of which are connected by edges. Their elementary structure makes graphs widely applicable, but the theoretical understanding of graphs is far from complete. Extremal graph theory aims to find connections between global parameters and substructure. A key topic is how a large average or minimum degree of a graph can force certain subgraphs (where the degree is the number of edges at a vertex). For instance, Erdős and Gallai proved in the 1960's that any graph of average degree at least $k$ contains a path of length $k$. Some of the most intriguing open questions in this area concern trees (connected graphs without cycles) as subgraphs. For instance, can one substitute the path from the previous paragraph with a tree? We will give an overview of open problems and recent results in this area, as well as their possible extensions to hypergraphs.